Evanescent microwave microscopy probe and methodology

ABSTRACT

An evanescent microwave microscopy probe substantially as described in the above specification and in the accompanying drawings including one or more of the novel features described in the above specification and drawings.

This application hereby claims priority to U.S. Provisional PatentApplication No. 60/620,592 filed on Oct. 20, 2004.

BACKGROUND OF THE INVENTION

The theoretical model for the change in resonant frequency of theresonator assembly as a function of the complex permittivity ofmaterials and the probe-sample geometry has been described. In contrastto existing theoretical description, the method of the present inventionis independent of electrical properties of the material, and applies todielectrics, conductors and superconductors. The method of the presentinvention is more general than prior methods. This generality isachieved by using perturbation theory imposed on electric field in thevicinity of the probe-tip. Prior methods assumed calculations based oncapacitance due to the gap between the spherical conducting tip andperfect conducting surface of the sample. Reaction of resonator probe onthe electric field existing in the gap and the sample does not leadnecessarily to results predicted by the prior methods. In order toachieve their results from our theory, we need to restrict our model byimposing additional condition on the reaction of the resonator probe onthe fields existing in the area outside the tip. Namely, thecoefficients in (9) and (10) should be the same (A′=A) to get theirresults. The advantage of this assumption gives a smooth transitionbetween insulators and ideal conductors by assuming b=1 in (8). Thephysics of superconductors are studied at the quantum level, but themacroscopic properties of the material from which it is derived must beconsistent within the classical theory of electromagnetics. The theoryand analysis proposed here allows the solution of the classicalelectrodynamic boundary value problem concerning a superconductormodeled as a dielectric with a large, negative real part for the complexpermittivity, which can be associated with the persistent current.

Prior work in this area used a shunt series combination. The maximum Qis solely determined by resistance of the series R-L-C probe equivalentcircuit and tuning network. However, sapphire capacitors have anintrinsic equivalent series resistance (ESR). The present inventionachieves substantially higher Q values than that of the prior art byarranging the sapphire tuning capacitors in parallel. By doing so theresistance is cut by 50% compared to a single shunt capacitor.Accordingly, this results in very high Q values and correspondingly highsensitivity.

BRIEF SUMMARY OF THE INVENTION

The present invention relates to near field microscopy and, moreparticularly to an evanescent microwave microscopy probe for use in nearfield microscopy and methodology for investigating the complexpermittivity of a material through evanescent microwave technology. Theprobe comprises a low loss, apertured, coaxial resonator that may betuned over a large bandwidth by a parallel shunt sapphire tuningnetwork. The transmission line of the probe utilizes high gradeparaffin, offering relatively low loss tangent and a very closedielectric match within the line. A chemically sharpened tip extendsslightly past the end aperture of the probe and emits a purelyevanescent field. This sensor is extremely sensitive, achieving Q valuesin excess of 0.5×10⁶ and a spatial resolution of 1.0×10⁻⁶ meters.

The physical construction of the probe according to the presentinvention dictates a purely evanescent field emanating from its tip. Asa result, in the context of use in quantitative microscopy, it is notnecessary to provide additional hardware and methodology to separate apropagative component from the field. The probe also allows an extremelylow loss impedance match to standardized equipment. The low loss coaxialresonator of the present invention theoretically has an infinitebandwidth but is practically governed by the constraints of physicallength and source bandwidth. The evanescent mode bandwidth is controlledby the aperture diameter, which is quite large compared with state ofthe art designs. The probe of the present invention also utilizes ashunt capacitive tuning network characterized by a low equivalent seriesresistance. As a result, the probe of the present invention, providesfor large resonant frequency selection range and extremely high Qvalues.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic representation of a cross-section of a probe inaccordance with the present invention,

FIG. 2 is a block diagram of a microscope in accordance with the presentinvention,

FIG. 3 is a diagram of the probe and coupling network,

FIG. 4 is a diagram showing the method of images,

FIG. 5 is a scanning electron micrograph of a superconducting filmhaving two distinct regions,

FIG. 6 is a plot of susceptibility loss versus temperature for asuperconducting film,

FIG. 7 is a pair of plots of resonant frequency versus distance betweenthe probe tip and the sample, wherein the data is collected at 79.4 Kand 298 K,

FIG. 8 is a plot showing the change in Q for the superconducting film at79.4 K,

FIG. 9 is a photograph of an embodiment of the microwave microscopyapparatus of the present invention,

FIG. 10 is a photograph of Ti—Au lines etched on sapphire at 20×magnification,

FIG. 11 is a plot of the change in Q,

FIG. 12 is a plot of the change in reflection coefficient images,

FIG. 13 is a circuit diagram representing the probe connected to asuperconductor,

FIG. 14 is a plot showing the change in Q for a superconducting film injunction area of 6° bi-crystal

FIG. 15 is a plot showing the tuned resonance with the probe tip onemicron above the SrTiO₃ crystal sample at 300 K,

FIG. 16 is a plot showing the frequency-shifted resonance with the probetip about 1 micron from the SrTiO₃ crystal sample at 302 K,

DETAILED DESCRIPTION OF THE INVENTION

The present invention generally relates to a microwave probe formicrowave microscopy and a method of using the same for generating highquality microwave data. More particularly, the apparatus and method ofthe present invention can be used to take high-precision, low-noise,measurements of material parameters such as permittivity, permeability,and conductively.

The probe can be used for the characterization of local electromagneticproperties of materials. The resonator-intrinsic, spatial resolution isexperimentally demonstrated herein. A first-order estimation of thesensitivity related to the probe tip-sample interaction for conductors,dielectrics, and superconductors is provided. An estimation of thesensitivity inherent to the resonant probe is presented. The probe issensitive in the range of theoretically estimated values, and hasmicrometer-scale resolution.

Probe Theory of Operation

In the field of evanescent microwave microscopy, the tip of the probeoperates in close proximity of the sample, where the tip radius andeffective field distribution range are much smaller than the resonatorexcitation wavelength. The propagating field exciting resonance in theprobe can be ignored and the probe tip-sample interaction can be treatedas quasi-static. This can be used for localized measurements and imageswith resolved features governed essentially by the characteristic sizeof the tip. The field distribution from the probe tip extends outward ashort distance, and as a material is entered into the near field of thetip, it will, interact with the evanescent field, perturbing theresonance of the probe. This perturbation is linked to the resonantstructure of the probe through the air gap coupling capacitance C_(C)between the tip and the material. This results in the loading of theresonant probe and alters the resonant frequency f_(r), quality factorQ, and reflection coefficient S₁₁ of the resonator.

If the air gap distance from tip to sample is held constant, the f_(r),Q, and S₁₁ variations related to the microwave properties of the samplecan be mapped as the probe tip is scanned over the sample. The microwaveproperties of a material are functions of permittivity ε, permeabilityμ, and conductivity σ.

Basic Probe Structure

Referring to FIG. 1, the microwave probe 10 of the present invention canbe constructed from a standard 0.085″ semi-ridged coaxial transmissionline. The probe 10 is based on an end-wall aperture coaxial transmissionline, where the resonator behaves as a series resonant circuit for oddmultiples of λ/4.

In constructing the probe 10, the center conductor is removed along withthe poly(tetrafluoroethylene) insulator and replaced with high purityparaffin 14. However, the invention is not restricted to paraffin andalternative materials can be used. For example, alternative materialswithin the scope of the present invention include, without limitation,magnesium oxide, titanium oxide, boron nitride, aluminas, and variousorganic polymeric materials.

Fashioning the probe 10 according to the foregoing paragraph results ina coaxial wave guide probe 10 rather that an open cavity. A copperaperture, having a thickness of about 0.010″, is soldered inside theouter shield 15, creating an end-wall aperture 12. A chemicallysharpened tip 17 is mounted on the center conductor 16 and electroplatedwith silver. The transmission line resonator is then reconstructed bycasting the sharpened, plated, center conductor 16 inside the outershield 15 with high purity paraffin 14. A short section of the originalpoly(tetrafluoroethylene) shielding replaces the paraffin 14 at thesharpened end of the coax, and is located directly above the end-wallaperture 12. This poly(tetrafluoroethylene) plug 18 is used to maintaintip-aperture alignment. The sharpened point 17 of the center conductor16 extends beyond the shielded end-wall aperture 12 of the resonator byapproximately 0.001″ or less. The purely evanescent probing field isradiated from the sharpened tip 17. In this manner, as the centerconductor 16 radius decreases, the spatial resolution of the probeincreases due to localization of the interaction between the tip 17 andsample 20.

Referring to FIG. 2, the microwave excitation frequency of the resonantprobe 10 can be varied in the network analyzer 40 bandwidth from 1 to 40GHz and is tuned by external capacitors 30. As is further illustrated inFIG. 3, the microscope probe can be coupled to the network analyzer 40through tuning network capacitors C₁ 31 and C₂ 32, which are connectedto the center conductor 16 and to the outer shield 15.

A block diagram of the microwave microscopy system is shown in FIG. 2.The changes in the probe's resonant frequency, quality factor (Q), andreflection coefficient are tracked by a Hewlett-Packard 8722ES networkanalyzer 40 through S₁₁ port measurements, as the probe 10 moves abovethe sample surface 20. The microwave excitation frequency of theresonant probe 10 can be varied within the bandwidth of the networkanalyzer 40 and is tuned to critical coupling by the tuning assembly 30.The tuning assembly 30 comprises two variable 2.5 to 8 pF capacitors 31,32. The tuning network has one capacitor C₁ 31 connected in-line withthe center conductor 16, and the other capacitor C₂ 32 is connected fromcenter conductor 16 to ground.

The X-Y axis stage 70 is driven by Coherent® optical encoded DC linearactuators. The probe 10 is frame-mounted to a Z-axis linear actuatorassembly and the height at which the probe 10 is above the sample 20 canbe precisely set. The X-Y stage actuators, network analyzer 40, and dataacquisition and collection are controlled by the computer 50. Theprogram that interfaces to the X-Y stage actuators, serial portcommunications, 8722ES GPIB interface, and data acquisition is writtenin National Instruments Labview® software. The complete evanescentmicrowave scanning system is mounted on a vibration-dampening table (seeFIG. 9).

According to one embodiment of the present invention, the externaltuning capacitor assembly 30 consists of two thermally compensatedsapphire capacitors in a shunt configuration. If a shunt is placed nearthe end of the resonator then the Q of the resonator will theoreticallyapproach infinity. Sapphire capacitors are advantageous because theyexhibit frequency invariance up to approximately 10 GHz. The capacitors31, 32 are preferably variable from, for example, about 4.5 to 8.0Picofarads. The position of the capacitors 31, 32 in the tuning assembly30 is optimized to reduce interaction. Shielding techniques may also beemployed to limit external interaction and leakage.

Mathematical Model and Methodology

As is noted above, the present invention also relates to methodology forinvestigating the complex permittivity of a material through evanescentmicrowave technology. More particularly, the methodology taught hereinis a scheme for investigating the complex permittivity of a material,independent of its electrical properties, through evanescent microwavespectroscopy.

The extraction of quantitative data through evanescent microwavemicroscopy requires a detailed configuration of the field outside theprobe-tip region. The solution of this field will clearly relate theperturbed signal to the probe tip-sample distance and physical materialproperties. It is essential that the mode of the field generated at thetip be evanescent, since mixed mode consisting of evanescent andpropagative will prevent quantitative measurements. The propagativewave's contribution to the tip-sample signal depends on the electricalproperties of the sample, and limits the resolution of the microscopysensor.

In analyzing conductors quantitatively the probe tip can be modeled as aconducting sphere and the sample as an ideal conductor. The tip andsample separation represents a capacitor with capacitance C_(c),resulting in a resonant frequency shift that is proportional to thevariation in C_(c). When a conducting material is placed near the tip aninteraction will cause charge and field redistribution. The method ofimages can be applied to model this redistribution of the field andrequires a series iteration of two image charges. This variation of thetip-sample capacitance results in a shift of the resonant frequency ofthe resonator.

To quantitatively analyze dielectric materials, an analysisincorporating the method of images can be applied. Also, the resonatortip is represented as a charged conducting sphere with potential V₀ andwhen closely placed over a dielectric material the dielectric will bepolarized by the electric field. This dielectric reaction to the tipcauses a redistribution of charge on the tip in order to maintain theequipotential surface of the sphere and also results in a shift infrequency of the resonator. Applying the method of images to model thefield redistribution requires a series of three image charges in aniterative process to meet boundary conditions at probe tip and thedielectric sample surface.

In this unified approach, perturbation theory for microwave resonatorsis applied dealing only with the field distribution outside the tip. Theexpression for the resonant frequency shift due to the presence of amaterial is $\begin{matrix}{{\frac{\Delta\quad f}{f_{0}} = {{- \frac{\int_{V}{\left\lbrack {{\left( {\Delta\quad ɛ} \right)\left( {\overset{\_}{E} \cdot {\overset{\_}{E}}_{0}} \right)} + {\left( {\Delta\quad\mu} \right)\left( {\overset{\_}{H} \cdot {\overset{\_}{H}}_{0}} \right)}} \right\rbrack\quad{\mathbb{d}V}}}{\int_{V}{\left( {{ɛ_{0}{\overset{\_}{E}}_{0}^{2}} + {\mu_{0}{\overset{\_}{H}}_{0}^{2}}} \right)\quad{\mathbb{d}V}}}} = \frac{f - f_{0}}{f_{0}}}},} & (1)\end{matrix}$where {overscore (E)} and {overscore (H)} are the perturbed fields, V isthe volume of a region outside the resonator tip, f is the resonantfrequency and f₀ is the reference frequency. The unperturbed field isgiven by $\begin{matrix}{{{E_{0}\left( {r,z} \right)} = {\frac{q}{4{\pi ɛ}_{0}}\frac{\left\lbrack {{r\hat{r}} + {\left( {z + {a_{1}^{\prime}r_{0}}} \right)\hat{z}}} \right\rbrack}{\left\lbrack {r^{2} + \left( {z + {a_{1}^{\prime}r_{0}}} \right)^{2}} \right\rbrack^{3/2}}}},{{{\overset{\_}{H}}_{0}} = {\sqrt{\frac{ɛ_{0}}{\mu_{0}}}{{\overset{\_}{E}}_{0}}}}} & (2)\end{matrix}$wherea′ ₁ =r ₀ +g  (3)with radius r₀ of the spherical tip and g as the gap between the tip andsurface of the sample. The potential V₀ on the spherical tip is given by$\begin{matrix}{V_{0} = {\frac{q}{4{\pi ɛ}_{0}r_{0}}.}} & (4)\end{matrix}$

By using the method of images (see FIG. 4), the perturbed electric fieldin the tip-sample region and the sample volume (where r₀ is much smallerthan the sample thickness) can be derived as $\begin{matrix}{{{{\overset{\_}{E}}_{1}\left( {r,z} \right)} = {\frac{q}{4{\pi ɛ}_{0}}{\sum\limits_{n = 1}^{\infty}{q_{n}\left\{ {\frac{\left\lbrack {{r\hat{r}} + {\left( {z + {a_{n}^{\prime}r_{0}}} \right)\hat{z}}} \right\rbrack}{\left\lbrack {r^{2} + \left( {z + {a_{n}^{\prime}r_{0}}} \right)^{2}} \right\rbrack^{3/2}} - {b\frac{\left\lbrack {{r\hat{r}} + {\left( {z - {a_{n}^{\prime}r_{0}}} \right)\hat{z}}} \right\rbrack}{\left\lbrack {r^{2} + \left( {z - {a_{n}^{\prime}r}} \right)^{2}} \right\rbrack^{3/2}}}} \right\}}}}},\quad{{{\overset{\_}{H}}_{1}} = {\sqrt{\frac{ɛ_{0}}{\mu_{0}}}{{\overset{\_}{E}}_{1}}}},} & (5) \\{{{{\overset{\_}{E}}_{2}\left( {r,z} \right)} = {\frac{1}{2{\pi\left( {ɛ + ɛ_{0}} \right)}}{\sum\limits_{n = 1}^{\infty}{q_{n}\frac{\left\lbrack {{r\hat{r}} + {\left( {z + {a_{n}^{\prime}r_{0}}} \right)\hat{z}}} \right\rbrack}{\left\lbrack {r^{2} + \left( {z + {a_{n}^{\prime}r_{0}}} \right)^{2}} \right\rbrack^{3/2}}}}}},{{{\overset{\_}{H}}_{2}} = {\sqrt{\frac{ɛ}{\mu}}{{\overset{\_}{E}}_{2}}}},} & (6)\end{matrix}$where μ is real and $\begin{matrix}{{a_{n}^{\prime} = {a_{1}^{\prime} - \frac{1}{a_{1}^{\prime} + a_{n - 1}^{\prime}}}},{q_{n} = {t_{n}q}},{t_{n} = \frac{{bt}_{n - 1}}{a_{1}^{\prime} + a_{n - 1}^{\prime}}},{t_{1} = 1},{b = \frac{ɛ - ɛ_{0}}{ɛ + ɛ_{0}}},{ɛ = {ɛ^{\prime} + {{\mathbb{i}ɛ}^{''}.}}}} & (7)\end{matrix}$

Importantly, for a tip in free space ε=ε₀ and μ=μ₀ at the location r=0and z=−g−r₀, {overscore (E)}₀={overscore (E)}₁={overscore (E)}₂ and{overscore (H)}₀={overscore (H)}₁={overscore (H)}₂, confirming theasymptotic behavior in (2), (5), and (6). By integrating the unperturbedelectric field in (2) and the perturbed electric fields in (5) and (6)over a region V outside the spherical tip the frequency shift (1)becomes $\begin{matrix}{{\left( \frac{\Delta\quad f}{f_{0}} \right)_{TOTAL} = {{\left( \frac{\Delta\quad f}{f_{0}} \right)_{1} + \left( \frac{\Delta\quad f}{f_{0}} \right)_{2}} = {{{- A}{\sum\limits_{n = 1}^{\infty}{t_{n}\left\{ {1 - {\frac{1}{2}\left( {1 - b} \right)\frac{1}{a_{1}^{\prime} + a_{n - 1}^{\prime}}}} \right\}}}} - {{A\left( \frac{\Delta\mu}{\Delta ɛ} \right)}\sqrt{\frac{ɛ}{\mu}}\sqrt{\frac{ɛ_{0}}{\mu_{0}}}{\sum\limits_{n = 1}^{\infty}{t_{n}\frac{b}{a_{1}^{\prime} + a_{n - 1}^{\prime}}}}}}}},\left( {A = A^{\prime}} \right),} & (8)\end{matrix}$where $\begin{matrix}{{\left( \frac{\Delta\quad f}{f_{0}} \right)_{1} = {{- A^{\prime}}{\sum\limits_{n = 1}^{\infty}{t_{n}\left\{ {1 - {\frac{1}{2}\left( {1 + b} \right)\frac{1}{a_{1}^{\prime} + a_{n - 1}^{\prime}}}} \right\}}}}},{{Reg}.\quad A},{{\Delta\mu} = {0\quad{and}}}} & (9) \\{{\left( \frac{\Delta\quad f}{f_{0}} \right)_{2} = {{- {A\left( {1 + {\frac{\Delta\mu}{\Delta ɛ}\sqrt{\frac{ɛ}{\mu}}\sqrt{\frac{ɛ_{0}}{\mu_{0}}}}} \right)}}{\sum\limits_{n = 1}^{\infty}{t_{n}\frac{b}{a_{1}^{\prime} + a_{n - 1}^{\prime}}}}}},{{Reg}.\quad B.}} & (10)\end{matrix}$

Parameters A and A′ are constants determined by the geometry of thetip-resonator assembly. Taking into account the real part of (8), we canfit this analytical expression, with our experimental data.

In one embodiment the method of the present invention is used to measurethe dielectric properties of the superconductor YBa₂Cu₃O_(7-δ). Asuperconductor can be treated as a dielectric material with a negativedielectric constant rather than a low loss conductor. In this embodimentthe probe 10 comprises a tuned, end-wall apertured coaxial transmissionline. The resonator probe 10 is coupled to a network analyzer 40 througha tuning network 30 and coupled to the sample 20 (see FIG. 2). When theresonator tip 12 is in close proximity to the sample 20, the resonator'sfrequency f will shift. In measuring the frequency shift, the proberesonant frequency reference is set at a fixed distance above thesample. This distance between probe tip 12 and sample 20 is sufficientto assure that the evanescent field emanating from the tip 12 will notinteract with the sample 20. The field dispersion from the probe tipextends outward a short distance with the amplitude of the evanescentfield decaying exponentially. As a sample 20 enters the near field ofthe probe 10, it will interact with the evanescent field, therebyperturbing it. This results in loading the resonator via the couplingand is considered part of the resonant circuit resulting in losses addedto the system, which decreases the microscope resonant frequency. Themeasured frequency shift versus tip-sample separation g generates atransfer function relating Δf to Δg, which is best fit with anelectrostatic field model generated from the method of images to extractthe complex permittivity values.

In a variation of the foregoing embodiment, the evanescent microwavemicroscopy system is adapted for making cryogenic measurements. Aminiature single-stage Joule-Thompson cryogenic system is fixed to theX-Y stage 70. The microwave probe is fitted through a bellows, whichprovides a vacuum seal and allows the probe to move freely over thesample, which is mounted on the cryogenic finger directly below theprobe.

In this embodiment, an YBa₂Cu₃O_(7-δ) superconducting thin film isfabricated by pulsed laser deposition. This deposition method results intwo distinct regions, 1 and 2, forming on a 0.5 mm thick LaAlO₃substrate (see FIG. 5). The superconductive transition temperatures forregion 1 and 2 of the film are T_(c)=92 K and 90 K respectively, whichare measured by plotting susceptibility loss versus temperature underdifferent amplitudes of alternating magnetic field at the frequency of 2MHz, as shown in FIG. 6. The measured frequency shift data is collectedfor both regions at 79.4 K and 298 K as shown in FIG. 7. Fittingparameters from (8) to our experimental data are consolidated in TableI. TABLE I SIMULATION FIT PARAMETERS FOR YBa₂Cu₃O_(7-□) SUPERCONDUCTINGTHIN FILM AT 79.4K AND 298K. A ε′/ε₀ r₀ μ/μ₀ REGIONS (10⁻⁴) (10⁸) ε″/ε₀(10⁻⁶ m) (10⁻⁴) REGION 1 at 79.4K 2.09 −9.2 −0.1 3.35 1 REGION 2 at79.4K 2.08 −9 −0.1 3.35 1 TRANSITION 2.08 −9.1 −0.1 3.35 1 REGION at79.4K REGION 1 at 298K 1.45 1 6.6 8 1 REGION 2 at 298K 1.45 1 6.85 8 1

Above the transition temperature (T_(c)), the superconductor behaveslike a metallic conductor, which changes the sign and magnitude of thereal and imaginary permittivity values (Table I). FIG. 7 shows thecurves from both regions below T_(c) and illustrates that there is adistinct measurable difference between these regions. The transitionsection connecting region 1 and 2 with the associated frequency shiftfit parameters generated at 79.4 K falls in between fit curves forregions 1 and 2. The model fit parameters for this transition segmentare A=2.08×10⁻⁴, which is the resonator scaling factor, the realcomponent of permittivity ε′=−9.13×10⁸ ε₀, the imaginary component ofpermittivity ε″=−0.1 ε₀, and the effective tip radius r₀=3.35 μm. FIG. 8shows a change in Q scan performed at 79.4 K over both regions andindicates the average dynamic range of Q in this scan between the twoareas is approximately 8000 , with the higher Q level associated withthe area of T_(c)=92 K and the lower Q level corresponding to region ofT_(c)=90 K.

System Resolution

The resolution of the probe is verified using a sapphire polycrystallinesubstrate with titanium-gold etched lines of widths ranging from 10 μmto 1 μm (see FIG. 10). The titanium is used to permit adhesion of thegold to the substrate and is approximately 100 nm thick, while thedeposition thickness of the gold is approximately 1 μm. The resonantfrequency of the probe is tuned to 2.67 GHz. The etched lines of thesample are scanned with the probe resulting in a change in frequency, Q,and magnitude of reflection plots.

The smallest physically resolvable feature for an evanescent probe isgoverned by the size of the tip radius, along with the height at whichthe tip is positioned above the feature. For example, to resolve a 5 μmphysical feature, the probe tip radius r₀ must be less than or equal to5 μm and should be no more than g=5 μm above it, where g is the distancefrom tip to sample.

The change in Q and change in magnitude of reflection coefficient imagesare illustrated in FIGS. 11 and 12, respectively. The data for theseplots are taken from a 20 μm×18 μm scan area around a 1 μm wide etchedline. The measured tip radius of the probe used is 1.2 μm with a standoff height of 2 μm and a 1 μm data acquisition step. The location of theetched line is indicated on each plot by arrows with correspondingmeasurements in micrometers. The one micrometer line was distinguishablein both plots, which gives the probe at least about 1 μm topographicalresolution. The Q values that are attainable with this tunable resonatorrange from 1.5×10⁴ to well over 10⁵. The dynamic range of the change inQ is approximately 5×10⁵ as shown in FIG. 11.

System Sensitivity

The Johnson noise limited sensitivity is analyzed for the presentinvention by setting the signal power equal to the noise power resultingin [(δε/ε)]=2.45×10⁻⁵.

The sensitivity of the evanescent microwave probe described here can beseparated into two categories. The first S_(r) is inherent to theresonator itself and directly proportional to it's quiescent operatingvalue Q. The other S_(f) is external to the resonator and solelydetermined by the tip-sample interaction. A noise threshold has to beconsidered in an evanescent microwave system, which also affectssensitivity.

The minimum detectable signal in an evanescent microwave microscopysystem has to be greater than the noise threshold created by theresonator probe, tuning network, and coupling to the sample. The noiseis generated by a resistance at an absolute temperature of T by therandom motion of electrons proportional to the temperature T within theresistor. This generates random voltage fluctuations at the resistorterminal, which has a zero average value, but a nonzero rms value givenby Planck's black body radiation law and can be calculated by theRaleigh-Jeans approximation [7] asV _(n(rms))=√{square root over (4kTBR)}  (11)where k=1.38×10⁻²³ J/K is Boltzmann's constant, T is the temperature inK, B is the bandwidth of the system in Hz, and R is the resistance in Ω.The resistance that results at critical coupling is the resistance Rthat produces noise in the system. Therefore, the signal level isrequired to be above this noise level for detection.Resonator Sensitivity S_(r)

The sensitivity approximation internal to the resonator S_(r) can bedetermined theoretically and experimentally. The theoretical value isanalytically approximated by considering the lumped series equivalentcircuit of the resonator, which has an inherent resonant frequency ω₀and Q associated with the lumped parameters R₀, L₀, and C₀. Thisconfiguration and associated parameters can be viewed as if the probetip is beyond the decay length of the evanescent field from a material,or in free space. If the probe tip is brought into close proximity andelectrically couples to the sample, the resonant frequency ω₀ and Q areperturbed to a new value ω′₀ and Q′, respectively, and are associatedwith new perturbed parameters R′₀, L′₀, and C′₀. The total impedancelooking into the terminals of the perturbed resonator coupled to asample can be written as $\begin{matrix}{Z_{TOTAL} = {{R_{0}^{\prime}\left\lbrack {1 + {j\quad{Q^{\prime}\left( {\frac{\omega}{\omega_{0}^{\prime}} - \frac{\omega_{0}^{\prime}}{\omega}} \right)}}} \right\rbrack}.}} & (12)\end{matrix}$

The magnitude of the reflection coefficient S₁₁ is related to Z_(TOTAL)by $\begin{matrix}{{S_{11} = \frac{Z_{TOTAL} - Z_{0}}{Z_{TOTAL} + Z_{0}}},} & (13)\end{matrix}$where Z₀ is the characteristic impedance of the resonant structure. Ifwe assume critical coupling, where the resonator is matched to thecharacteristic impedance of the feed transmission line at resonantfrequency, then R′₀≈Z₀ at ω≈ω′₀ and S_(r) is defined in [5] as$\begin{matrix}{{S_{r} = {\frac{\mathbb{d}S_{11}}{\mathbb{d}\omega} \approx {\frac{Q^{\prime}}{\omega_{0}^{\prime}}\left( {1 - \frac{\Delta\omega}{\omega_{0}^{\prime}}} \right)}}},} & (14)\end{matrix}$where Δω=ω−ω′₀.Probe Sensitivity S_(f)

The external sensitivity determined by tip-sample interaction of theresonator is based on a λ/4 section of transmission line, with thelumped parameter series equivalent circuit coupled to an equivalentcircuit model of a superconductor shown in FIG. 7. The series lumpedparameter circuit for the resonator consists of R₀, L₀, and C₀ and theprobe tip coupling to the superconductor is represented by C_(C). Theequivalent circuit model of the superconductor is comprised of R_(S),L_(S), C_(S), and L_(C), where the series combination of R_(S) and L_(S)represents the normal conduction. The element L_(C) signifies thekinetic inductance of the Cooper-pair flow and C_(S) is related todisplacement current. The superconductor equivalent circuit contains thenecessary circuit elements in the appropriate configuration to representnot only a superconductor, but a metallic conductor and a dielectric.

The equivalent circuit model for the probe coupled to a superconductoris illustrated in FIG. 7, where the equivalent circuit model for thesuperconductor is derived from the two- fluid model [8]. The lumpedcircuit representation of the superconductor consists of capacitanceC_(S), the inductance for normal carrier flow L_(S), and resistivityρ=1/σ₁ shunted by kinetic inductance L_(C)=1/ωσ₂. The parameters C_(S)and L_(S) are considered to have minimal effects [8] when thesuperconductor is subjected to low frequencies and is neglected in thisanalysis. The conductivity ratio y=σ₁/σ₂ is correlated to the impedanceratio y=ωL_(C)/ρ and in the limit of large y (y>>1), σ₂=0 and L_(C)>>1[8]. The opposite extreme, y<<1 results in L_(C) approaching 0, while σ₂advances toward infinity. The superconductive samples for this studywere subjected to a frequency of approximately 1 GHz and are of aninductive nature. The superconductor with an inductive nature hasL_(C)<<R_(S).

The impedance Z₁ is the parallel combination of R_(S) and L_(C) and isrepresented as $\begin{matrix}{Z_{1} = {\frac{{j\omega}\quad L_{C}R_{S}}{R_{S} + {{j\omega}\quad L_{C}}}.}} & (15)\end{matrix}$

The impedance Z₂ is the series combination of C_(C) and Z_(1,) whichresults in $\begin{matrix}{Z_{2} = {{\frac{1}{{j\omega}\quad C_{C}} + \frac{{j\omega}\quad L_{C}R_{S}}{R_{S} + {{j\omega}\quad L_{C}}}} = {\frac{R_{S} + {{j\omega}\quad L_{C}} + {{j\omega}\quad{C_{C}\left( {{j\omega}\quad L_{C}R_{S}} \right)}}}{{j\omega}\quad{C_{C}\left( {R_{S} + {{j\omega}\quad L_{C}}} \right)}}.}}} & (16)\end{matrix}$

The impedance Z₃ is the parallel combination of Z₂ and C₀ given by$\begin{matrix}{{\frac{1}{Z_{3}} = {{\frac{1}{Z_{2}} + {{j\omega}\quad C_{0}}} = {\frac{{j\omega}\quad{C_{C}\left( {R_{S} + {{j\omega}\quad L_{C}}} \right)}}{R_{S} + {{j\omega}\quad L_{C}} + {{j\omega}\quad{C_{C}\left( {{j\omega}\quad L_{C}R_{S}} \right)}}} + {j\omega C}_{0}}}}\begin{matrix}{Z_{3} = \frac{R_{S} - {\omega^{2}L_{C}C_{C}R_{S}} + {{j\omega}\quad L_{C}}}{{{j\omega}\quad{C_{C}\left( {R_{S} + {{j\omega}\quad L_{C}}} \right)}} + {C_{0}\left( {R_{S} - {\omega^{2}L_{C}C_{C}R_{S}} + {{j\omega}\quad L_{C}}} \right)}}} \\{= {- {\frac{j}{\omega}\left\lbrack \frac{R_{S} - {\omega^{2}L_{C}C_{C}R_{S}} + {{j\omega}\quad L_{C`}}}{{C_{C}\left( {R_{S} + {{j\omega}\quad L_{C}}} \right)} + {C_{0}\left( {R_{S} - {\omega^{2}L_{C}C_{C}R_{S}} + {{j\omega}\quad L_{C}}} \right)}} \right\rbrack}}} \\{= {{- \frac{j}{\omega}}{Z_{3}^{\prime}.}}}\end{matrix}} & (17)\end{matrix}$

The total impedance Z_(TOTAL) looking into the terminals of the probecoupled to a superconductor sample is$Z_{total} = {R_{0} + {{j\omega}\quad L_{0}} - {\frac{j}{\omega}{Z_{3}^{\prime}.}}}$

The complex impedance Z₃ can be represented as$Z_{3} = {{\frac{1}{j\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack} = {- {{\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack}.}}}$

At resonance, the inductive and capacitive reactances cancel; therefore,$\begin{matrix}{{{{{j\omega}\quad L_{0}} - {\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack}} = 0},{{\omega^{2}L_{0}} = {{{Re}\left( Z_{3}^{\prime} \right)}.}}} & (18)\end{matrix}$

This allows us to solve for perturbed frequency ω in terms of theperturbed lumped circuit parameters in an iterative process, where wewill be taking a first-order approximation. The combination of (7) and(8) results in $\begin{matrix}\begin{matrix}{{\omega^{2}L_{0}} = \frac{{R_{S}^{2}\left( {1 - {\omega^{2}L_{C}C_{C}}} \right)}\left( {C_{C} + C_{0} - {\omega^{2}C_{0}L_{C}C_{C}}} \right)}{{R_{S}^{2}\left( {C_{C} + C_{0} - {\omega^{2}C_{0}L_{C}C_{C}}} \right)}^{2} + {\omega^{2}{L_{C}^{2}\left( {C_{C} + C_{0}} \right)}^{2}}}} \\{= {\frac{1}{\left( {C_{C} + C_{0}} \right)}\frac{\left( {1 - {\omega^{2}L_{C}C_{C}} - {\omega^{2}L_{C}\frac{C_{0}C_{C}}{C_{C} + C_{0}}}} \right)}{\left( {1 - {2\omega^{2}L_{C}\frac{C_{0}C_{C}}{C_{C} + C_{0}}}} \right)}}} \\{= \frac{1}{\left( {C_{C} + C_{0}} \right)}} \\{\frac{1}{\left( {1 - {2\omega^{2}L_{C}\frac{C_{0}C_{C}}{C_{C} + C_{0}}}} \right)\left\lbrack {1 + {\omega^{2}L_{C}{C_{C}\left( {1 + \frac{C_{0}}{C_{C} + C_{0}}} \right)}}} \right\rbrack}} \\{= {\frac{1}{\left( {C_{C} + C_{0}} \right)}{\frac{1}{1 + {\omega^{2}L_{C}\frac{C_{C}^{2}}{C_{C} + C_{0}}}}.}}}\end{matrix} & (19)\end{matrix}$

Therefore, for the first iteration, we have the equation $\begin{matrix}{\omega_{0}^{\prime 2} = {\frac{1}{L_{0}\left( {C_{C} + C_{0}} \right)}{\frac{1}{\left\lbrack {1 + {\frac{L_{C}}{L_{0}}\left( \frac{C_{C}}{C_{C} + C_{0}} \right)^{2}}} \right\rbrack}.}}} & (20)\end{matrix}$

Solving for ω′₀ in (20) results in $\begin{matrix}{{\omega_{0}^{\prime} = {\omega_{0}\frac{1}{\sqrt{1 + \frac{C_{C}}{C_{0}}}}\frac{1}{\sqrt{1 + \frac{L_{C}}{L_{0}}}\left( \frac{C_{C}}{C_{C} + C_{0}} \right)^{2}}}},} & (21)\end{matrix}$Where $\frac{L_{C}}{L_{0}} ⪡ 1.$

The Taylor expansion of (21) gives $\begin{matrix}{\omega_{0}^{\prime} = {{{\omega_{0}\left( {1 - {\frac{1}{2}\frac{C_{C}}{C_{0}}}} \right)}\left\lbrack {1 - {\frac{1}{2}\frac{L_{C}}{L_{0}}\frac{C_{C}^{2}}{\left( {C_{0} + C_{C}} \right)^{2}}}} \right\rbrack}.}} & (22)\end{matrix}$

The sensitivity S_(f) for a superconductor is defined as $\begin{matrix}{{S_{f} = {\frac{g_{S}R_{S}^{2}}{2\quad\pi}{\frac{\mathbb{d}\omega_{0}^{\prime}}{\mathbb{d}L_{C}}}}},} & (23)\end{matrix}$where ${g_{S} = \frac{A_{eff}}{\lambda_{L}}},$A_(eff) is the effective tip area, and λ_(L) is the London penetrationdepth. Therefore, the sensitivity S_(f) for a superconductor is found bytaking the derivative of ω′₀ with respect to L_(C) in (22) and is givenby $\begin{matrix}{S_{f} = {\frac{g_{S}R_{S}^{2}}{2\quad\pi}\quad{{{\omega_{0}\left( {1 - \frac{C_{C}}{2C_{0}}} \right)}\left\lbrack {\frac{1}{\left( {2L_{0}} \right)}\frac{C_{C}^{2}}{\left( {C_{C} + C_{0}} \right)^{2}}} \right\rbrack}.}}} & (24)\end{matrix}$

The ability of the probe to differentiate between regions of differentconductivity within a superconductor Δσ/σ is defined as $\begin{matrix}{\frac{\Delta\quad\sigma}{\sigma} = {{\left( \frac{V_{n{({rms})}}}{V_{in}} \right)/S_{f}}S_{r}{\sigma.}}} & (25)\end{matrix}$

The probe couples to a metallic sample through the coupling capacitanceC_(C) and the conductor is represented as the series combination ofR_(S) and L_(S). An equivalent circuit of a metallic sample does notcontain the circuit elements L_(C) and C_(S) in the two-fluid equivalentcircuit (see FIG. 13). Therefore, C_(S)=0 and L_(C)=∞. The impedance Z₁is the series combination of C_(C), R_(S), and L_(S) and is representedas $\begin{matrix}\begin{matrix}{Z_{1} = {R_{S} + {j\quad\omega\quad L_{S}} + \frac{1}{j\quad\omega\quad C_{C}}}} \\{\quad{= {\frac{1 + {j\quad\omega\quad C_{C}R_{S}} - {\omega^{2}L_{S}C_{C}}}{j\quad\omega\quad C_{C}}.}}}\end{matrix} & (26)\end{matrix}$

The parallel combination of Z₁ and C₀ results in $\begin{matrix}{\frac{1}{Z_{2}} = {{j\quad\omega\quad C_{0}} + \frac{j\quad\omega\quad C_{C}}{\left( {1 - {\omega^{2}L_{S}C_{C}}} \right) + {j\quad\omega\quad C_{C}R_{S}}}}} & \quad \\{\quad{= \frac{{j\quad\omega\quad{C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}} + {j\quad\omega\quad C_{C}} - {\omega^{2}C_{0}C_{C}R_{S}}}{\left( {1 - {\omega^{2}L_{S}C_{C}}} \right) + {j\quad\omega\quad C_{C}R_{S}}}}} & \quad \\{\quad{{= \frac{j\quad{\omega\left\lbrack {C_{C} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)} - {\omega^{2}C_{0}C_{C}R_{S}}} \right\rbrack}}{1 - {\omega^{2}L_{S}C_{C}} + {j\quad\omega\quad C_{C}R_{S}}}},}} & \quad \\{{and}\quad{the}\quad{impedance}\quad Z_{2}\quad{is}} & \quad \\{Z_{2} = \frac{\left( {1 - {\omega^{2}L_{S}C_{C}}} \right) + {j\quad\omega\quad C_{C}R_{S}}}{j\quad{\omega\left\lbrack {{C_{C}{C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}} - {\omega^{2}C_{0}C_{C}R_{S}}} \right\rbrack}}} & (27) \\{\quad{= {{- \frac{j}{\omega}}{Z_{2}^{\prime}.}}}} & \quad\end{matrix}$

The total impedance Z_(TOTAL) looking into the terminals of the probecoupled to a conductor sample is$Z_{TOTAL} = {R_{0} + {j\quad\omega\quad L_{0}} - {\frac{j}{\omega}{Z_{2}^{\prime}.}}}$

The complex impedance Z₃ can be represented as $\begin{matrix}{Z_{2} = {\frac{1}{j\quad\omega}\left\lbrack {{Re}\left( Z_{2}^{\prime} \right)} \right\rbrack}} \\{\quad{= {- {{\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{2}^{\prime} \right)} \right\rbrack}.}}}}\end{matrix}$

At resonance, the inductive and capacitive reactance cancel; therefore,$\begin{matrix}\begin{matrix}{{{{j\quad\omega\quad L_{0}} - {\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{2}^{\prime} \right)} \right\rbrack}} = 0},} \\{{\omega^{2}L_{0}} = {{{Re}\left( Z_{2}^{\prime} \right)}.}}\end{matrix} & (28)\end{matrix}$

The impedance Z′₂ is represented as $\begin{matrix}{Z_{2}^{\prime} = {\frac{\left( {1 - {\omega^{2}L_{S}C_{C}}} \right) + {j\quad\omega\quad C_{C}R_{S}}}{\left\lbrack {C_{C} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)} + {j\quad\omega\quad C_{0}C_{C}R_{S}}} \right\rbrack}.}} & (29)\end{matrix}$

Taking the real part of (29), we have $\begin{matrix}\begin{matrix}{{{Re}\left( Z_{2}^{\prime} \right)} = \frac{{\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)\left\lbrack {C_{C} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}} \right\rbrack} + {\omega^{2}C_{0}C_{C}^{2}R_{S}^{2}}}{\left\lbrack {C_{C} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}} \right\rbrack^{2} + {\omega^{2}C_{0}^{2}C_{C}^{2}R_{S}^{2}}}} \\{= {\frac{{C_{C}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}^{2} + {\omega^{2}C_{0}C_{C}^{2}R_{S}^{2}}}{\left\lbrack {C_{C} + {C_{0}\left( {1 - {\omega^{2}L_{S}C_{C}}} \right)}} \right\rbrack^{2} + {\omega^{2}C_{0}^{2}C_{C}^{2}R_{S}^{2}}}.}}\end{matrix} & (30)\end{matrix}$

The numerator and denominator of (30) are considered separately, so thenumerator is expanded and results in(C _(C) +C ₀)−ω²(L _(S) C _(C) ²+2C ₀ L _(S) C _(C) −C ₀ C _(C) ² R _(S)²)+ω⁴ C ₀ C _(C) ² L _(S) ²  (31)

The ω⁴ term in (31) is discarded due to insignificance and thedenominator of (30) is expanded as(C _(C) +C ₀−ω² L _(S) C _(C) C ₀)²+ω² C ₀ ² C _(C) ² R _(S) ²=(C _(C)+C ₀)²−2ω² L _(S)(C _(C) +C ₀)C _(C) C ₀+ω⁴ C ₀ ² C _(C) ² L _(S) ²+ω² C₀ ² C _(C) ² R _(S) ²  (32)

Likewise, the ω⁴ term in (32) is neglected and the combination of (31)and (32) appear as $\begin{matrix}{\frac{\left( {C_{C} + C_{0}} \right) - {\omega^{2}\left( {{L_{S}C_{C}^{2}} + {2C_{0}L_{S}C_{C}} - {C_{0}C_{C}^{2}R_{S}^{2}}} \right)}}{\left( {C_{C} + C_{0}} \right)^{2} - {2\quad\omega^{2}{L_{S}\left( {C_{C} + C_{0}} \right)}\quad C_{0}C_{C}} + {\omega^{2}C_{0}^{2}C_{C}^{2}R_{S}^{2}}}.} & (33)\end{matrix}$

Factoring out (C_(C)+C₀) in numerator and denominator of (33) andsubstituting the result into (28) produces $\begin{matrix}{{\omega^{2}L_{0}} = {\frac{1}{\left( {C_{C} + C_{0}} \right)}{\frac{1 - {\omega^{2}\frac{\left( {{L_{S}C_{C}^{2}} + {2C_{0}L_{S}C_{C}} - {C_{0}C_{C}^{2}R_{S}^{2}}} \right)}{\left( {C_{C} + C_{0}} \right)}}}{1 - {2\omega^{2}\frac{L_{S}C_{C}C_{0}}{\left( {C_{C} + C_{0}} \right)}} + {\omega^{2}\frac{C_{0}^{2}C_{C}^{2}R_{S}^{2}}{\left( {C_{C} + C_{0}} \right)^{2}}}}.}}} & (34)\end{matrix}$

Reducing (34) and multiplying by${1 + {\omega^{2}\frac{\left\lbrack {{L_{S}{C_{C}\left( {C_{C} + {2C_{0}}} \right)}} - {C_{0}C_{C}^{2}R_{S}^{2}}} \right\rbrack}{\left( {C_{C} + C_{0}} \right)}}},$results in $\begin{matrix}{{\omega^{2}L_{0}} = {\frac{1}{\left( {C_{C} + C_{0}} \right)}{\frac{1}{1 + {\omega^{2}\frac{L_{S}C_{C}^{2}}{\left( {C_{C} + C_{0}} \right)}} + {\omega^{2}\frac{C_{0}C_{C}^{2}R_{S}^{2}}{\left( {C_{C} + C_{0}} \right)}\left( {\frac{C_{0}}{C_{C} + C_{0}} - 1} \right)}}.}}} & (35)\end{matrix}$

The relation ω₀ ²/(1+C_(C)/C₀) with ω₀ ²=1/L₀C₀ as a zero-orderapproximation to our iterative process is substituted into (35)producing a first-order approximation $\begin{matrix}{\omega_{0}^{\prime^{2}} = {\frac{1}{L_{0}\left( {C_{C} + C_{0}} \right)}{\frac{1}{1 + {\frac{L_{S}}{L_{0}}\left( \frac{C_{C}}{C_{C} + C_{0}} \right)^{2}} - {\frac{C_{0}R_{S}^{2}}{L_{0}}\left( \frac{C_{C}}{C_{C} + C_{0}} \right)^{3}}}.}}} & (36)\end{matrix}$

Rewriting (36) and taking the square root of both sides and neglectinghigher-order terms, we have the first-order approximation for theperturbed resonant frequency due to the coupling of the probe to aconductor. $\begin{matrix}{\omega_{0}^{\prime} = {\omega_{0}\frac{1}{\sqrt{1 + \frac{C_{C}}{C_{0}}}}{\frac{1}{\sqrt{1 + {\frac{L_{S}}{L_{0}}\left( \frac{C_{C}}{C_{C} + C_{0}} \right)^{2}}}}.}}} & (37)\end{matrix}$

The Taylor expansion of (37) gives $\begin{matrix}{\omega_{0}^{\prime} = {{{\omega_{0}\left( {1 - \frac{C_{C}}{2C_{0}}} \right)}\left\lbrack {1 - {\frac{L_{S}}{2L_{0}}\frac{C_{C}^{2}}{\left( {C_{C} + C_{0}} \right)^{2}}}} \right\rbrack}.}} & (38)\end{matrix}$

The sensitivity S_(f) for a conductor is defined as $\begin{matrix}{{S_{f} = {\frac{g_{S}R_{S}^{2}}{2\pi}{\frac{\mathbb{d}\omega_{0}^{\prime}}{\mathbb{d}L_{S}}}}},} & (39)\end{matrix}$where ${g_{S} = \frac{A_{eff}}{\delta}},$A_(eff) is the effective tip area, and δ is the skin depth. Therefore,the sensitivity S_(f) (39) for a conductor is found by taking thederivative of ω′₀ with respect to L_(S) in (38) and results in$\begin{matrix}{S_{f} = {\frac{g_{S}R_{S}^{2}}{2\quad\pi}{{{\omega_{0}\left( {1 - \frac{C_{C}}{2\quad C_{0}}} \right)}\left\lbrack {\frac{1}{2\quad L_{0}}\frac{C_{C}^{2}}{\left( {C_{C} + C_{0}} \right)^{2}}} \right\rbrack}.}}} & (40)\end{matrix}$

The ability of the probe to differentiate between regions of differentconductivity Δσ/σ is defined as $\begin{matrix}{{\frac{\Delta\quad\sigma}{\sigma} = {{\left( \frac{V_{n{({rms})}}}{V_{in}} \right)/S_{f}}S_{r}\sigma}},} & (31)\end{matrix}$where v_(n(rms)) is given in (11) and v_(in) is the probe input voltage.

The probe also couples to a dielectric sample through the couplingcapacitance C_(C) and the dielectric is represented as the parallelcombination of R_(S) and C_(S). The equivalent circuit of an insulatingsample does not contain the circuit elements L_(C) and L_(S) from thetwo-fluid equivalent circuit. Therefore, L_(S)=0 and L_(C)=∞. Theimpedance Z₁ is the parallel combination of R_(S) and C_(S) and isrepresented as $\begin{matrix}{Z_{1} = {\frac{R_{S}}{{{j\omega}\quad C_{S}R_{S}} + 1}.}} & (42)\end{matrix}$

The series combination of Z₁ and C_(C) result in $\begin{matrix}{Z_{2} = {{\frac{1}{j\quad\omega\quad C_{C}} + \frac{1}{{j\quad\omega\quad C_{S}R_{S}} + 1}} = {\frac{1 + {{j\omega}\quad C_{S}R_{S}} + {{j\omega}\quad C_{C}R_{S}}}{{j\omega}\quad{C_{C}\left( {{{j\omega}\quad C_{S}R_{S}} + 1} \right)}}.}}} & (43)\end{matrix}$

The impedance Z₃ is the parallel combination of Z₂ and C₀ and isrepresented as $\begin{matrix}{{\frac{1}{Z_{3}} = {\frac{{j\omega}\quad{C_{C}\left( {1 + {{j\omega}\quad C_{S}R_{S}}} \right)}}{\left( {1 + {{j\omega}\quad C_{C}R_{S}} + {{j\omega}\quad C_{S}R_{S}}} \right)} + {{j\omega}\quad C_{0}}}},{Z_{3} = {\frac{1 + {{j\omega}\quad C_{C}R_{S}} + {{j\omega}\quad C_{S}R_{S}}}{{{j\omega}\quad{C_{C}\left( {1 + {{j\omega}\quad C_{S}R_{S}}} \right)}} + {{j\omega}\quad{C_{0}\left( {1 + {{j\omega}\quad C_{C}R_{S}} + {{j\omega}\quad C_{S}R_{S}}} \right)}}} = {{- \frac{j}{\omega}}{Z_{3}^{\prime}.}}}}} & (44)\end{matrix}$

The total impedance Z_(TOTAL) looking into the terminals of the probecoupled to a dielectric sample is$Z_{TOTAL} = {R_{0} + {{j\omega}\quad L_{0}} - {\frac{j}{\omega}{Z_{3}^{\prime}.}}}$

The complex impedance Z₃ can be represented as$Z_{3} = {{\frac{1}{j\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack} = {- {{\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack}.}}}$

At resonance, the inductive and capacitive reactance cancel; hence,$\begin{matrix}{{{{{j\omega}\quad L_{0}} - {\frac{j}{\omega}\left\lbrack {{Re}\left( Z_{3}^{\prime} \right)} \right\rbrack}} = 0},{{\omega^{2}L_{0}} = {{{Re}\left( Z_{3}^{\prime} \right)}.}}} & (45)\end{matrix}$

The quantity jωR_(S) is factored out in the numerator and denominator of(44) and the result is placed into (45), giving $\begin{matrix}{{\omega^{2}L_{0}} = {{Re}\left\{ \frac{1 + {{j\omega}\quad{R_{S}\left( {C_{C} + C_{S}} \right)}}}{\left( {C_{C} + C_{0}} \right) + {{j\omega}\quad{R_{S}\left\lbrack {{C_{C}C_{S}} + {C_{0}\left( {C_{C} + C_{S}} \right)}} \right\rbrack}}} \right\}}} \\{= {\frac{\left( {C_{C} + C_{0}} \right) + {\omega^{2}{{R_{S}^{2}\left( {C_{C} + C_{S}} \right)}\left\lbrack {{C_{C}C_{S}} + {C_{0}\left( {C_{C} + C_{S}} \right)}} \right\rbrack}}}{\left( {C_{C} + C_{0}} \right)^{2} + {\omega^{2}{R_{S}^{2}\left\lbrack {{C_{C}C_{S}} + {C_{0}\left( {C_{C} + C_{S}} \right)}} \right\rbrack}^{2}}}.}}\end{matrix}$

R_(S) is neglected since it is large, so $\begin{matrix}{{\omega^{2}L_{0}} \approx \frac{\left( {C_{C} + C_{S}} \right)}{{C_{C}C_{S}} + {C_{0}\left( {C_{C} + C_{S}} \right)}}} \\{= {\frac{1}{C_{0}}{\frac{1}{\left\lbrack {1 + \frac{C_{C}C_{S}}{C_{0}\left( {C_{C} + C_{S}} \right)}} \right\rbrack}.}}}\end{matrix}$Therefore, $\begin{matrix}{\omega_{0}^{\prime^{2}} = {\frac{1}{L_{0}C_{0}}{\frac{1}{\left\lbrack {1 + \frac{C_{C}C_{S}}{C_{0}\left( {C_{C} + C_{S}} \right)}} \right\rbrack}.}}} & (46)\end{matrix}$

Solving for ω′₀ in (46) results in $\begin{matrix}{\omega_{0}^{\prime} = {\omega_{0}{\frac{1}{\sqrt{1 + \frac{C_{C}C_{S}}{C_{0}\left( {C_{C} + C_{S}} \right)}}}.}}} & (47)\end{matrix}$

The Taylor expansion of (47) gives $\begin{matrix}{\omega_{0}^{\prime} = {{\omega_{0}\left\lbrack {1 - \frac{C_{C}C_{S}}{2{C_{0}\left( {C_{C} + C_{S}} \right)}}} \right\rbrack}.}} & (48)\end{matrix}$

The sensitivity S_(f) for a dielectric is defined as $\begin{matrix}{{S_{f} = {\frac{g_{S}}{2\pi}{\frac{\mathbb{d}\omega_{0}^{\prime}}{\mathbb{d}C_{S}}}}},} & (49)\end{matrix}$where ${g_{S} = \frac{A_{eff}}{\xi_{S}}},$A_(eff) is the effective tip area, and ξ_(s) is the decay length of theevanescent wave, which is approximately 100 μm. Therefore, thesensitivity S_(f) for a dielectric is found by taking the derivative ofω′₀ with respect to C_(S) in (48) $\begin{matrix}{S_{f} = {\frac{g_{S}\omega_{0}}{4\pi}{\frac{C_{C}^{2}}{{C_{0}\left( {C_{C} + C_{S}} \right)}^{2}}.}}} & (50)\end{matrix}$

The ability of the probe to differentiate between regions of differentpermittivity Δε/ε is defined as $\begin{matrix}{\frac{\Delta\quad ɛ}{ɛ} = {{\left( \frac{V_{n\quad{({rms})}}}{V_{in}} \right)/S_{f}}S_{r}{ɛ.}}} & (51)\end{matrix}$

The experimental verification of the sensitivity for superconductors isperformed on a YBa₂Cu₃O₇₋₆₇ coated SrTiO₃ bi-crystal of 60° orientationmismatch. Resonant frequency shift measurements are taken, resulting incomplex permittivity values for two separate locations below T_(c) at79.4 K. The measurements are taken in the boundary at points C and Dshown in FIG. 8. The sensitivities given by (14), (24), and (25) arelisted in Table II. TABLE II SENSITIVITY AND ASSOCIATED PARAMETERS FORSUPERCONDUCTORS ε′/ε₀ (10⁸) S_(r) S_(f) Δσ/σ Position C −8.94 9.03 ×10⁻⁶ 1.13 × 10⁻⁶ 1.0 × 10⁻² Position D −8.87 1.04 × 10⁻⁵ 1.13 × 10⁻⁶ 8.6× 10⁻³

The sensitivity parameters comprise C_(C)=1.36×10⁻¹⁵ F, C₀=8.91×10⁻¹² F,L₀=2.03×10⁻⁸ H, R_(S)=1×10⁻⁶ ΩQ, σ=3.3×10⁸ S/m, and g_(s)=1.02×10⁻³ Theexperimental results show that Δσ/σ≅7.8×10⁻³.

The experimental verification of the sensitivity for conductors is alsoperformed on the YBa₂Cu₃O_(7-δ) coated SrTiO₃ bi-crystal of 60°orientation mismatch. The measurements are taken at the same locationsfor the superconductor sensitivity, in the boundary at points C and D(FIG. 14) at a temperature of 300 K. The sensitivities given by (14),(40), and (41) are listed in Table III. The sensitivity parametersconsist of C_(C)=1.36×10⁻¹⁵ F, C₀=8.91×10⁻¹² F, L₀=2.03×10⁻⁸ H,R_(S)=7.76×10⁻⁴ Ω[8], σ=1.28×10³ S/m, and g_(c)=1.54×10⁻⁴. Theexperimental results have shown that Δσ/σ≅2.4×10⁻². TABLE IIISENSITIVITY AND ASSOCIATED PARAMETERS FOR CONDUCTORS ε″/ε₀ S_(r) S_(f)Δσ/σ Position C 6.3 6.83 × 10⁻⁶ 5.9 × 10⁻² 8.36 × 10⁻² Position D 6.155.95 × 10⁻⁶ 5.9 × 10⁻² 9.91 × 10⁻²

The experimental verification of the sensitivity for dielectrics isperformed on single crystal SrTiO₃ utilizing the ferroelectricdependence on temperature property of the material, i.e., ε_(r)=f(7).The probe tip is set to a 1 μm distance above the sample and tuned to aresonant frequency of 1.114787 GHz at a temperature of 300 K and isillustrated in FIG. 15. The temperature is raised in 0.2 K incrementsuntil the resonance shifted in frequency to 1.114792 GHz at 302 K due tothe change in dielectric constant and is shown in FIG. 16. The change indielectric constant is determined using the Curie-Weiss law and resultsin Δε/ε≅6.23×10⁻³. The sensitivity parameters consist of ε′/ε₀=320.8,C_(C)=1.36×10⁻¹⁵ F, C₀=8.91×10⁻¹² F, C_(S)=4.37×10⁻¹⁵ F, andg_(s)=1.54×10⁻⁶. The lowest theoretically estimated change inpermittivity that can be detected by the sensor was Δε/ε=5.75×10⁻⁴.

It is noted that terms like “preferably,” “commonly,” and “typically”are not utilized herein to limit the scope of the claimed invention orto imply that certain features are critical, essential, or evenimportant to the structure or function of the claimed invention. Rather,these terms are merely intended to highlight alternative or additionalfeatures that may or may not be utilized in a particular embodiment ofthe present invention.

Having described the invention in detail and by reference to specificembodiments thereof, it will be apparent that modifications andvariations are possible without departing from the scope of theinvention defined in the appended claims. More specifically, althoughsome aspects of the present invention are identified herein as preferredor particularly advantageous, it is contemplated that the presentinvention is not necessarily limited to these preferred aspects of theinvention.

1. An evanescent microwave microscopy probe substantially as describedin the above specification and in the accompanying drawings includingone or more of the novel features described in the above specificationand drawings.
 2. An evanescent microwave microscopy probe comprising: adielectric support member, and a conductor transmission line comprisingat least one electrically isolated conductive element extending alongthe length of said dielectric support member and forming a tapered probetip; an electrically conductive sheath mounted on said dielectricsupport member said sheath enclosing said electrically isolatedconductive element and forming a wave guide.
 3. A method ofinvestigating the complex permittivity of a material through evanescentmicrowave technology as described in the above specification and in theaccompanying drawings including one or more of the novel featuresdescribed in the above specification and drawings.